Weaker bound allowing zero eigenvalues

Prove that every graph G of order n satisfies 2n ≤ (n−n^+(G))(n−n^+(G)+3).

Background

Because the main conjecture may be difficult to prove, the authors propose a weaker conjecture involving the number of non-positive adjacency eigenvalues, n−n^+(G). They note that this formulation is equivalent to the main conjecture when the adjacency matrix has no zero eigenvalues and that it refines a question attributed to Mohar.